Suppose gcd(m,n) = 1.

Let d be a common divisor of mn and (m+n). Then there exist x, y s.t.

xd = mn and yd = m+n.

xd = m(yd - m) or (my - x)d = m^2. And similarly,

xd = (yd - n)n or (ny - x)d = n^2.

So d divides m^2 and n^2,

d is a common divisor of m and n,

d must be 1,

gcd(mn, m+n) = 1.

Hope this helps.